Austin L Buchanan

Oklahoma State University | Oklahoma State · School of Industrial Engineering and Management

Consider the task of dividing a state into k contiguous political districts whose populations must not differ by more than one person, following current practice for congressional districting in the USA. A widely held belief among districting experts is that this task requires at least k − 1 county splits. This statement has appeared in expert test...

This paper studies the problems of partitioning the vertices of a graph G = (V, E) into (or covering with) a minimum number of low-diameter clusters from the lenses of approximation algorithms and integer programming. Here, the low-diameter criterion is formalized by an s-club, which is a subset of vertices whose induced subgraph has diameter at mo...

In the academic literature and in expert testimony, the Polsby-Popper score is the most popular way to measure the compactness of a political district. Given a district with area A and perimeter P, its Polsby-Popper score is given by (4πA)/P^2. This score takes values between zero and one, with circular districts achieving a perfect score of one. I...

The most successful approaches for the TSP use the integer programming model proposed in 1954 by Dantzig, Fulkerson, and Johnson (DFJ). Although this model has exponentially many subtour elimination constraints (SECs), it has been observed that relatively few of them are needed to prove optimality in practice. This leads us to wonder: What is the c...

Political Districting to Minimize County Splits When dividing a state into districts for elections, one traditional criterion is that political subdivisions like counties and cities should not be divided unnecessarily. Some states go as far as to say that the number of county splits should be minimized, but previously there was no scalable exact me...

Motivated by applications in political districting, we consider the task of partitioning the n vertices of a planar graph into k connected components. We propose an extended formulation for this task that has two desirable properties: (i) it uses just O(n) variables, constraints, and nonzeros, and (ii) it is perfect. To explore its ability to solve...

This mini survey covers optimization, heuristic, and sampling methods for political districting. It appears as a chapter in the Encyclopedia of Optimization (Springer, 3rd edition) edited by Panos M. Pardalos and Oleg A. Prokopyev.

When partitioning a state into political districts, a common criterion is that political subdivisions like counties should not be split across multiple districts. This criterion is encoded into most state constitutions and is sometimes enforced quite strictly by the courts. However, map drawers, courts, and the public typically do not know what amo...

Cliques and their generalizations are frequently used to model “tightly knit” clusters in graphs and identifying such clusters is a popular technique used in graph-based data mining. One such model is the s-club, which is a vertex subset that induces a subgraph of diameter at most s. This model has found use in a variety of fields because low-diame...

When constructing political districting plans, prominent criteria include population balance, contiguity, and compactness. The compactness of a districting plan, which is often judged by the “eyeball test”, has been quantified in many ways, e.g., Length-Width, Polsby-Popper, and Moment-of-Inertia. This paper considers the number of cut edges, which...

In critical node problems, the task is to identify a small subset of so-called critical nodes whose deletion maximally degrades a network’s “connectivity” (however that is measured). Problems of this type have been widely studied, for example, for limiting the spread of infectious diseases. However, existing approaches for solving them have typical...

Beginning in the 1960s, techniques from operations research began to be used to generate political districting plans. A classical example is the integer programming model of Hess et al. (Operations Research 13(6):998--1006, 1965). Due to the model's compactness-seeking objective, it tends to generate contiguous or nearly-contiguous districts, altho...

The usual integer programming formulation for the maximum clique problem has several undesirable properties, including a weak LP relaxation, a quadratic number of constraints and nonzeros when applied to sparse graphs, and poor guarantees on the number of branch-and-bound nodes needed to solve it. With this as motivation, we propose new mixed integ...

Cliques and their generalizations are frequently used to model ``tightly knit'' clusters in graphs, and identifying such clusters is a popular technique used in graph-based data mining. One such model is the $s$-club, which is a vertex subset that induces a subgraph of diameter at most $s$. This model has found use in a variety of fields because lo...

The celebrated Motzkin–Straus formulation for the maximum clique problem provides a nontrivial characterization of the clique number of a graph in terms of the maximum value of a nonconvex quadratic function over a standard simplex. It was originally developed as a way of proving Turán’s theorem in graph theory, but was later used to develop compet...

Jose L. Walteros and Austin Buchanan

We study statistical calibration, i.e., adjusting features of a computational model that are not observable or controllable in its associated physical system. We focus on functional calibration, which arises in many manufacturing processes where the unobservable features, called calibration variables, are a function of the input variables. A major...

In the analysis of networks, one often searches for tightly knit clusters. One property of a “good” cluster is a small diameter (say, bounded by k), which leads to the concept of a k-club. In this paper, we propose new path-like and cut-like integer programming formulations for detecting these low-diameter subgraphs. They simplify, generalize, and/...

In the article “A linear‐size zero‐one programming model for the minimum spanning tree problem in planar graphs” (Networks 39(1) (2002), 53‐60), Williams introduced an extended formulation for the spanning tree polytope of a planar graph. This formulation is remarkably small (using only O(n) variables and constraints) and remarkably strong (definin...

This article considers the node‐weighted Steiner tree (NWST) problem and the maximum‐weight connected subgraph (MWCS) problem, which have applications in the design of telecommunication networks and the analysis of biological networks. Exact algorithms with provable worst‐case runtimes are provided. The first algorithm for NWST runs in time for n‐v...

In many network applications, one searches for a connected subset of vertices that exhibits other desirable properties. To this end, this paper studies the connected subgraph polytope of a graph, which is the convex hull of subsets of vertices that induce a connected subgraph. Much of our work is devoted to the study of two nontrivial classes of va...

The vertex cover polytopes of graphs do not admit polynomial-size extended formulations. This motivates the search for polyhedral analogues to approximation algorithms and fixed-parameter tractable (FPT) algorithms. In this paper, we take the FPT approach and study the k-vertex cover polytope (the convex hull of vertex covers of size k). Our main r...

In this article, a heuristic is said to be provably best if, assuming , no other heuristic always finds a better solution (when one exists). This extends the usual notion of “best possible” approximation algorithms to include a larger class of heuristics. We illustrate the idea on several problems that are somewhat stylized versions of real-life ne...

This paper explores techniques for solving the maximum clique and vertex coloring problems on very largescale real-life networks. Because of the size of such networks and the intractability of the considered problems, previously developed exact algorithms may not be directly applicable. The proposed approaches aim to reduce the network instances to...

This paper considers the minimum k -connected d -dominating set problem, which is a fault-tolerant generalization of the minimum connected dominating set ( MCDS) problem. Three integer programming formulations based on vertex cuts are proposed ( depending on whether d < k, d D k, or d > k) and their integer hulls are studied. The separation problem...

A connected dominating set (CDS) is commonly used to model a virtual backbone of a wireless network. To bound the distance that information must travel through the network, we explicitly restrict the diameter of a CDS to be no more than s leading to the concept of a dominating s -club. We prove that for any fixed positive integer s it is NP-complet...

We describe an algorithm for the maximum clique problem that is parameterized by the graph’s degeneracy $d$ . The algorithm runs in $O\left( nm+n T_d \right) $ time, where $T_d$ is the time to solve the maximum clique problem in an arbitrary graph on $d$ vertices. The best bound as of now is $T_d=O(2^{d/4})$ by Robson. This shows tha...